We propose a new approach for computing tunneling rates in quantum or thermal field theory with multiple scalar fields. It is based on exact analytical solutions of piecewise linear potentials with many segments that describes any given potential to arbitrary precision. The method is first developed for the single field case in 3 and 4 space-time dimensions and demonstrated on examples of classical potentials as well as the calculation of quantum fluctuations. A systematic expansion of the potential beyond the linear order is considered, taking into account higher order corrections, which paves the way for multiple scalar fields. We thereby provide a fast semi-analytical tool for evaluating the bounce action for theories with an extended scalar sector.

15 pages, 12 figures, polygonal bounces extended to multi-fields, computation of the prefactor, added references